Present address: Division of Biology, Imperial College London at Silwood Park, Ascot, Berkshire SL5 7PY, United Kingdom.
Wright's adaptive topography describes gene frequency evolution as a maximization of mean fitness in a constant environment. I extended this to a fluctuating environment by unifying theories of stochastic demography and fluctuating selection, assuming small or moderate fluctuations in demographic rates with a stationary distribution, and weak selection among the types. The demography of a large population, composed of haploid genotypes at a single locus or normally distributed phenotypes, can then be approximated as a diffusion process and transformed to produce the dynamics of population size, N, and gene frequency, p, or mean phenotype, . The expected evolution of p or is a product of genetic variability and the gradient of the long-run growth rate of the population, , with respect to p or . This shows that the expected evolution maximizes , the mean Malthusian fitness in the average environment minus half the environmental variance in population growth rate. Thus, as a function of p or represents an adaptive topography that, despite environmental fluctuations, does not change with time. The haploid model is dominated by environmental stochasticity, so the expected maximization is not realized. Different constraints on quantitative genetic variability, and stabilizing selection in the average environment, allow evolution of the mean phenotype to undergo a stochastic maximization of . Although the expected evolution maximizes the long-run growth rate of the population, for a genotype or phenotype the long-run growth rate is not a valid measure of fitness in a fluctuating environment. The haploid and quantitative character models both reveal that the expected relative fitness of a type is its Malthusian fitness in the average environment minus the environmental covariance between its growth rate and that of the population.
Sewall Wright's adaptive topography is a central paradigm in evolutionary biology, demonstrating that, under certain assumptions, gene frequency evolution in a constant environment maximizes the mean fitness of individuals in a population (Wright 1932, 1937, 1969). On Wright's adaptive topography a population is located at a point on the surface of mean fitness as a function of gene frequencies. Assuming genotypic fitnesses constant in time, with random mating and approximate linkage equilibrium among loci, natural selection in a constant environment causes gene frequencies to change such that the population moves uphill on the adaptive topography, increasing the mean fitness in the population. Fisher expressed a similar concept in his “fundamental theorem of natural selection,” in which the rate of increase of the mean Malthusian fitness, or population growth rate, equals the additive genetic variance in fitness (Fisher 1930, 1958 chap. 2; interpretations by different authors reviewed by Crow 2002). Wright's initial intuitive explanations of the adaptive topography, and his shifting balance theory of evolution (Wright 1931, 1932), were later clarified by explicit models demonstrating multiple peaks on the adaptive topography, and showing that gene frequency change by selection depends on the gradient of mean fitness with respect to gene frequencies (Wright 1935a,b, 1937, 1939, 1956, 1969).
Adaptive topographies have been criticized, most notably by Fisher (1941) and Moran (1964) who produced examples of evolution in which mean fitness either does not change or decreases. Fisher's example concerned a plant population in which an allele causing self-fertilization (without reducing pollen output or seed production) becomes rapidly fixed. This entails that genotypic fitnesses are not constant but change with the genotypic frequencies (Porcher and Lande 2005). Moran's example concerned a negative epistatic interaction between two polymorphic loci in a population initially in complete linkage disequilibrium, containing only the two most-fit genotypes, which by random mating and recombination then produce less-fit genotypes, reducing the mean fitness without changing gene frequencies. These examples respectively violated Wright's assumptions of random mating with constant genotypic fitnesses, and approximate linkage equilibrium. Wright (1942, 1949) showed that under nonrandom mating, or with frequency-dependent fitnesses, evolution generally does not maximize mean fitness. Kimura (1965) developed the concept of quasi-linkage equilibrium, demonstrating that when the strength of epistatic selection is much less than the recombination rate, random mating quickly produces a nearly constant linkage disequilibrium after which gene frequency evolution increases the mean fitness. The main criticisms were reviewed and answered by Wright (1967, 1988) and Crow and Kimura (1970, pp. 195–236).
Wright's adaptive topography therefore remains an important conceptual tool for analysis and understanding of evolution in both simple and complex genetic systems (e.g., Simpson 1953; Futuyma 1998; Gavrilets 2004). An analogous adaptive topography was derived for evolution of the mean phenotypes of quantitative characters, influenced by multiple genes and environmental effects, assuming constant phenotypic fitnesses and (multivariate) normal distributions of phenotypes and additive genetic values, with approximately constant phenotypic and additive genetic variances (and covariances) (Lande 1976, 1979, 1982).
The theory reviewed above concerns evolution in a constant environment. However, all populations experience temporally fluctuating environments, causing population size and natural selection to fluctuate. Analyzing empirical time series of selection coefficients, using existing methods for measuring selection, is a natural way of describing fluctuating selection. A population in a fluctuating environment is conceived as evolving toward a continually shifting optimum, with the mean fitness often temporarily decreasing (Crow and Kimura 1970, p. 236; Gillespie 1991, p. 305). Environmental change is one component of Fisher's (1958, pp. 44–45) “deterioration of the environment.” This raises the question: Is there a general principle of adaptation in a fluctuating environment?
A related question in stochastic demography concerns how to quantify the long-run growth rate of population size under density-independent growth in a fluctuating environment. Assuming a stationary distribution of environmental states, fundamental demographic results have been obtained using the concept of the “long-run growth rate” defined as the expected rate of increase of ln N, the natural logarithm of population size (Cohen 1977, 1979). In general, for populations with overlapping generations, the long-run growth rate can be approximated as , in which r is the population growth rate in the average environment, and σ2e is the environmental variance in population growth rate (Tuljapurkar 1982). The term long-run growth rate arises from the following consideration. In the absence of density dependence, the stochastic process for ln N has constant incremental mean and variance per unit time, describing Brownian motion with a deterministic trend (Lande and Orzack 1988). Starting from ln N(0) at time t= 0, at a later time the distribution of ln N(t) is asymptotically normal with mean and variance σ2et. The slope of a population trajectory on the log scale, [ln N(t) − ln N(0)]/t, then has mean and variance σ2e/t. With increasing time the variance of the slope approaches zero and all populations eventually have the same slope, (Cohen 1977, 1979; Tuljapurkar 1982; Caswell 2001; Lande et al. 2003).
Early models of fluctuating selection on a single genetic locus in a population with discrete nonoverlapping generations showed that the final outcome of selection is determined by the geometric mean fitness of a genotype through time (Dempster 1955; Haldane and Jayakar 1963; Felsenstein 1976; Gillespie 1973, 1991, p. 147). The analog of the geometric mean fitness in continuous time is the long-run growth rate, which is also applicable with overlapping generations (Tuljapurkar 1982; Lande and Orzack 1988; Caswell 2001; Lande et al. 2003). For alleles at a single haploid locus (or an asexual population) the genotype with the highest long-run growth rate will eventually become fixed in the population (Gillespie 1991, p. 147). At a diploid locus, the long-run growth rate of a rare heterozygote determines whether it can successfully invade a large population in a fluctuating environment (Tuljapurkar 1982; Gillespie 1991; Charlesworth 1994), and a permanent fluctuating polymorphism of two alleles can be maintained if the long-run growth rate of the heterozygote exceeds that of either heterozygote (Gillespie 1973, 1991). At a multiallelic diploid locus, the long-run growth rates of all the genotypes determine the eventual outcome of weak fluctuating selection (Turelli 1981). However, before accepting the geometric mean fitness or the long-run growth rate as the expected fitness of a genotype (Gillespie 1973, 1977; Tuljapurkar 1982, 1990; Metz et al. 1992; Caswell 2001, chap. 14.6; Doak et al. 2005), note that the above conclusions concern the eventual fate of a gene, and not the expected change in gene frequency over a short period of time, by which the expected selection coefficients and expected fitnesses of genotypes should be defined.
Here I analyze a well-known model of fluctuating selection on a haploid locus, and develop an analogous model of fluctuating selection on quantitative characters. In both models the expected evolution is governed by the pattern of genetic variation and the gradient of the long-run growth rate of the population, , with respect to gene frequency, p, or mean phenotype, . This shows that the expected evolution increases . The long-run growth rate therefore provides an adaptive topography determining the expected selection, which does not change with time despite environmental fluctuations. The population, represented as a point on this surface, is expected to evolve as a stochastic maximization of . In a haploid population this principle is not realized because the expected optimizing selection is dominated by stochasticity. For a quantitative character a stochastic maximization of can occur under fluctuating selection if the variance of the standardized selection gradient (in units of phenotypic standard deviations) is small or the heritability is low, which often will be satisfied. The adaptive surface as a function of , therefore provides an intuitive guide to phenotypic evolution in a fluctuating environment.
The analysis also reveals the form of the expected relative fitness of a genotype or phenotype within a population. Based on the results of simple genetic models mentioned above, prevailing opinion plausibly suggests that genotypic or phenotypic fitnesses should be measured by long-run growth rates. In contrast, for both the haploid (asexual) and quantitative character models of fluctuating selection, I find that the expected relative fitness of a genotype or phenotype is its Malthusian fitness in the average environment minus the covariance between its growth rate and that of the population.
Several simplifying assumptions are required for accuracy of the present results. Population size is assumed to be sufficiently large to neglect random genetic drift, so that all stochasticity arises from environmental fluctuations. States of the environment affecting the vital rates of survival and reproduction are assumed to have a stationary distribution with no (or short-term) serial correlation. Diffusion approximations developed here require that selection should be weak and density-independent. Under weak selection, Malthusian fitnesses provide an accurate description of evolution in continuous time models approximating natural selection in discrete time (Charlesworth 1994). Accuracy of demographic projections using the long-run growth rate of a population also requires small or moderate fluctuations in the vital rates (Tuljapurkar 1982, 1990; Lande and Orzack 1988; Caswell 2001; Lande et al. 2003, 2006).
Population density that identically decrements the Malthusian fitness of each genotype maintains their relative fitnesses and describes density-independent selection (Crow and Kimura 1970, chap. 1; Prout 1980; Lande and Shannon 1996). In an age-structured population, density dependence may act on any or all of the age-specific survival or fecundity rates. Charlesworth (1994, chap. 3) notes that, in a density-independent population, changing the fecundity and/or juvenile survival rates of all genotypes by the same proportion does not alter relative Malthusian fitnesses; whereas in a population maintained near a constant size by density dependence, changing the survival rate of all genotypes during their reproductive ages does not alter relative Malthusian fitnesses. The present models therefore assume that demographic density dependence acts on these suites of vital rates in the life history, in concert with fluctuations in population size, to produce fluctuating density-independent selection.
For quantitative characters the evolutionary dynamics of the mean phenotype derived here assume normality of the phenotypic and additive genetic distributions. Extension of these formulas for multiple generations in an adaptive topography requires that the (co)variances of these distributions remain constant, which should be approximately correct if selection is weak and linkage is not tight among loci with epistatic fitness interactions (Bulmer 1971, 1976; Lande 1982).
Continuously varying, quantitative characters, such as body size and shape, usually are influenced by multiple genes and environmental effects. Such characters often have an approximately normal phenotype distribution when measured on an appropriate scale, such as the natural logarithm of the raw measure (Wright 1968, chaps. 10 and 11; Falconer and MacKay 1996, chap. 17; Lynch and Walsh 1998, chap. 11). Breeding values (net additive genetic effects) for the characters also tend to be normally distributed when multiple genetic loci in linkage equilibrium produce phenotypic effects of comparable magnitude, because of the Central Limit Theorem of statistics (Kendall and Stuart 1977, chap. 7.26). Even with linkage disequilibrium caused by strong phenotypic selection, evolution of the mean phenotype is accurately predicted by normal distribution theory (Turelli and Barton 1994). In artificial selection experiments, the mean phenotype often undergoes substantial changes in response to selection while the additive genetic variance remains relatively stable, largely due to genetic segregation and recombination (Wright 1977, chaps. 7 and 8; Crow and Kimura 1970, pp. 236–239; Falconer and MacKay 1996). A small input of new additive genetic variability by mutation and immigration may therefore suffice to maintain the additive genetic (co)variance(s) nearly constant. The taxonomic and temporal scales over which phenotypic and additive genetic (co)variances evolve are subjects of empirical investigation (Turelli 1988; Phillips and Arnold 1999a,b; Roff 2000; Steppan et al. 2002).
Finally, all genotypes are assumed to have the same developmental plasticity in response to environmental change, so there is no genotype–environment interaction. If developmental plasticity is partially adaptive, as commonly observed (Schmalhausen 1949, pp. 7–10; Via and Lande 1985; Schlichting and Pigliucci 1998, chaps. 3 and 9), this would effectively reduce the intensity of fluctuations in selection. For simplicity, the present model describes the evolutionary dynamics of the mean phenotype measured in a common reference environment (e.g., the average environment) after subtracting from the observed time series of mean phenotypes the developmental responses to environmental change (see Appendix).
Let ni be the number of the ith allele at a single locus in a large haploid population, or the number of the ith genotype in a large asexual population, and denote the vector of genotypic numbers as n= (n1, n2, …). In a random environment with a stationary distribution and no serial correlation, the stochastic growth rate of each genotype can be approximated as a diffusion process, characterized by the first two infinitesimal moments. The infinitesimal mean is the expected change in ni per unit time, and the infinitesimal (co)variance is the (co)variance of the change in ni (and nj) per unit time, given n (Karlin and Taylor 1981, chap. 15; Gillespie 1991, chap. 4; Lande et al. 2003, chap. 2),
Here ri is the Malthusian fitness, or intrinsic rate of increase (at low population density), of genotype i in the average environment, and g(n) is a density-dependent function common to all genotypes. The environmental covariance between the growth rates of genotypes i and j is cij. Short-term serial correlation of the environment can be incorporated into the diffusion, approximately, by integrating cij across all time lags (Turelli 1977; Lande et al. 2003).
The process can be analyzed on the natural log scale using the transformation formulas for a diffusion process (Karlin and Taylor 1981, p. 173; Gillespie 1991, p. 157). The infinitesimal mean of ln ni, and the corresponding infinitesimal variances and covariances, are
where is the constant “long-run growth rate,” or stochastic intrinsic rate of increase, of a large density-independent population composed purely of genotype i in a random environment.
Now consider two haploid genotypes, or more generally one particular genotype versus all others. The stochastic demography of genotypic numbers (eq. 1) can be transformed to the natural log of total population size, ln N= ln (n1+n2), and the gene frequency, p=n1/(n1+n2). Using the transformation formulas for a bivariate diffusion process (Gillespie 1991; Engen et al. 2005), the infinitesimal mean and variance of ln N are
The density-independent long-run growth rate of the population, , at a given gene frequency is the mean Malthusian fitness in the average environment, r=pr1+ (1 −p)r2, minus half the environmental variance in population growth rate, σ2e=p2c11+ 2p(1 −p)c12+ (1 −p)2c22.
Here the expected relative fitness of the ith genotype, , equals its Malthusian fitness in the average environment minus the environmental covariance between its growth rate and that of the population. The infinitesimal variance of gene frequency is
Finally, the infinitesimal covariance between ln N and p is
The same formulas (eqs. 3–5) apply for multiple genotypes, provided that the partial derivatives with respect to p are taken holding constant the relative frequencies of the other genotypes (Wright 1969; Crow and Kimura 1970). Because the expected relative fitness of a genotype, , enters all of these equations, particularly the expected change in gene frequency (eq. 4a), only as its difference among genotypes, an identical arbitrary quantity may be added to the expected relative fitness of each genotype, depending on the desired population mean.
The dynamics of gene frequency (eqs. 4a,b) constitute an autonomous system independent of population size. In extensive form equations (4a,b) are identical to the diffusion approximation for the classical model of fluctuating selection on a haploid population (Gillespie 1991, chap. 4.2). In contrast, the population dynamics depend on the gene frequency (eq. 3), and changes in population size are also correlated with evolutionary changes through the infinitesimal covariance between ln N and p (eq. 5).
In the haploid model the mean Malthusian fitness in the average environment, r, is a linear function of gene frequency, whereas the environmental variance of the population, σ2e, is a quadratic function of gene frequency. With environmental variance an intermediate optimum may exist because the long-run growth rate generally has negative curvature, defined as
The inequality arises because the result is the variance of the difference between the genotypes in demographic responses to environmental fluctuations, which must be nonnegative. It is somewhat counterintuitive that the long-run growth rate of the population, , always exceeds the weighted mean long-run growth rate of the genotypes, , unless the difference vanishes when γ= 0, for example all genotypes respond identically to the environment (all cij=c), or there is no polymorphism, p= 0 or 1 (see Fig. 1).
The expected selection gradient, the partial derivative of the long-run growth rate, , with respect to the allele frequency, p (eq. 4a), shows that the expected evolution is always in the direction of increasing . However, this maximum principle is not realized in a haploid population because stochasticity overwhelms the expected optimizing selection. This can be seen from a simple analysis of the stochastic stability around an optimal gene frequency, , corresponding to a maximum of the long-run growth rate, rmax. An intermediate optimum, with , exists when c12−c22 < r1−r2 < c11−c12. Because is a quadratic function of p it can be represented exactly by the Taylor expansion . The absolute curvature γ measures the strength of stabilizing selection around the optimum, which also determines the expected rate of return to . Thus, when the environmental covariance between the genotypes is less than their mean environmental variance, fluctuating selection acts to stabilize the gene frequency. If this tendency is stronger than the stochasticity causing it to diverge from the stable equilibrium , a quasi-stationary distribution of gene frequency will be concentrated in a narrow region around the equilibrium, where the infinitesimal mean is approximately linear, (eq. 4a, definitions after eq. 3), and the infinitesimal variance of p is approximately constant, (eq. 4b). Around an optimum the quasi-stationary distribution (or sojourn time distribution) (Karlin and Taylor 1981) has the same shape as Wright's (1931, 1969) stationary distribution, proportional to . Substituting M and V, noting that γ cancels from the ratio, gives an approximately normal distribution for the gene frequency with mean and variance . However, in a distribution of frequencies with mean the largest possible variance is . This indicates that the quasi-stationary distribution cannot be concentrated around and explains why fluctuating selection cannot maintain polymorphism at a haploid locus (Gillespie 1991, chap. 4.2).
Figure 1 illustrates that the gene frequency tends to undergo large fluctuations, sometimes repeatedly shifting between near fixation and near loss of one genotype, until eventual quasi-fixation of the genotype with the highest long-run growth rate.
Diploid inheritance complicates this basic model. However, for one locus with additive fitness effects the gene frequency obeys the same stochastic dynamics as in the haploid model but multiplied by 1/2 (Gillespie 1991, chap. 4.4), so an adaptive topography for fluctuating selection exists also in that case.
Let z denote a quantitative phenotypic character, or a vector of correlated characters, for an individual. The phenotype distribution, p(z), is assumed to be normal with mean , which may evolve, and variance (or covariance matrix) P. The distribution of breeding values (net additive genetic effects) also is assumed to be normal with mean and variance (or covariance matrix) G. The mean breeding value, and hence the mean phenotype, may evolve in response to phenotypic selection, but the phenotypic and additive genetic variances (or covariance matrices) are assumed to remain approximately constant. The phenotypic (co)variance (matrix) equals the sum of the additive genetic (co)variance (matrix) plus the environmental (co)variance (matrix), E, which includes the following sources of phenotypic variation among individuals: developmental noise, microenvironmental effects on individual development, and nonadditive genetic effects.
Define the number of individuals with phenotypes in the range z to z+dz as n(z)dz. The total population size is and the frequency of phenotype z is p(z) =n(z)/N. Let n(z) have infinitesimal mean [r(z) −g(n)]n(z) and let the infinitesimal covariance between n(y) and n(z) be c(y, z)n(y)n(z). Here r(z) is the Malthusian fitness of individuals with phenotype z in the average environment, and g(n) denotes a density-dependent function of the distribution of genotypic numbers, n(z) =Np(z). The infinitesimal mean and variance of N are then, respectively, [r−g(n)]N and σ2eN2 where
The environmental variance in growth rate of the population, σ2e, is the mean environmental covariance in growth rate among the phenotypes. This is always nonnegative because it is a (continuous) quadratic form based on covariance function c(y, z), which must be non-negative definite.
Applying the natural log transformation to the process for N (Karlin and Taylor 1981), the infinitesimal mean and infinitesimal variance of ln N are obtained,
where is the long-run growth rate of the population with a given mean phenotype, the mean Malthusian growth rate in the average environment minus half the environmental variance in population growth rate.
To derive the infinitesimal mean and variance of the mean phenotype, the genetic response to selection is expressed as a product of factors involving phenotypic selection within a generation and the genetic response to selection across generations. Assuming that phenotypes and breeding values are normally distributed, the regression of breeding values on phenotypes is linear and homoscedastic. The evolutionary response to selection at a particular time t can then be represented using the phenotypic selection differential, S(t), the change in mean phenotype due to selection within a generation, accounting for individual survival and fecundity but not inheritance, or in terms of the selection gradient, β(t), the regression of individual relative fitness on the individual phenotypes (Lande 1979, 1982; Lande and Arnold 1983),
in which the superscript “−1” denotes matrix inverse. The infinitesimal mean and variance of can therefore be expressed as
where and Var [β(t)]=P−1 Var [S(t)]P−1 are, respectively, the mean and the (co)variance (matrix) of the selection gradient, given a particular value of the mean phenotype but averaged over the distribution of environmental states. Evolution of the mean phenotype is an autonomous system not influenced by population dynamics.
The mean and variance of the selection differential, S(t), can be derived by analyzing selection within a generation, including individual survival and reproduction but not inheritance. In a constant environment the phenotypic Malthusian fitnesses are applied to the phenotype frequencies over an infinitesimal time interval to calculate the phenotypic effect of selection (Lande 1982). In the present model with continually fluctuating selection, I apply the stochastic growth rates to the phenotypes to evaluate the influence of phenotypic selection. The mean phenotype before selection is . Phenotype frequencies after selection within a generation are p*(z) =p(z) +dp*(z). Given the infinitesimal change in the mean phenotype after selection is and the mean and variance of the selection differential are
The superscript “T” denotes transposition in the case of multiple characters.
From the infinitesimal means and covariances of the phenotypic numbers, the transformation formulas for a diffusion, or direct calculations as in Gillespie (1991, pp. 150–151), produce the infinitesimal means and infinitesimal covariances of the phenotype frequencies after selection within a generation,
where . The mean Malthusian fitness in the average environment, r, and the environmental variance of the population, σ2e, are defined in equation (6a).
Substituting (9a) into (8a) yields the expected selection differential,
Here is the expected relative fitness of an individual with phenotype z, its Malthusian fitness in the average environment minus the environmental covariance between its growth rate and that of the population.
The expected evolution of the mean phenotype can also be expressed in an alternate form, as the gradient of the long-run growth rate with respect to the mean phenotype, . Because the normal phenotype distribution, p(z), is a function of , we can differentiate with respect to , and using gives the selection gradient
where is the expected relative fitness in equation (10a). Formulas (7b) and (10a,b) show that at any given the expected selection gradient equals the gradient of the long-run growth rate, . Thus, the expected evolution of the mean phenotype in a fluctuating environment obeys the gradient dynamics
The expected selection gradient, , at a given gives the steepest uphill direction on the surface of the long-run growth rate of the population, , but the expected direction of evolution is modified by the pattern of heritable variation embodied in the G matrix. The expected evolution nevertheless increases the long-run growth rate because along this trajectory
Of course, stochasticity in selection can temporarily decrease the long-run growth rate of the population by causing fluctuations around the expected trajectory.
Substituting (9b) into (8b), and rearranging, yields
because y and z are independent, . Thus, the variance of the selection differential measures an aspect of the curvature of the environmental covariance function between phenotypes (Lande and Arnold 1983).
Finally, the infinitesimal covariance between ln N and the mean phenotype is
FLUCTUATING OPTIMUM PHENOTYPE
For example consider a model of fluctuating selection on a single quantitative character, assuming that the Malthusian fitness function retains the same shape as in the average environment, r(z), but fluctuates horizontally through time (Slatkin and Lande 1976; Bull 1987; Lande and Shannon 1996). Vertical fluctuation of the Malthusian fitness function would increase environmental stochasticity in population dynamics without affecting selection and evolution. Denote an optimum phenotype at time t as θ(t) and denote its mean and variance as and σ2θ. Taylor expansion of the Malthusian fitness function yields
Employing this in (6a) produces the environmental variance of the population,
The integral is the average slope of the Malthusian fitness function in the neighborhood of the phenotype distribution (the selection gradient for a given mean phenotype) in the average environment, β (Lande and Arnold 1983). The environmental variance of the population is minimized when the mean phenotype is at the optimum in the average environment, β= 0. Environmental fluctuations produce an additional force of stabilizing selection through the contribution to the expected selection gradient from the term (eq. 10c). Substituting (13a) into (11), integrating by parts using a property of the normal phenotype distribution (above 10b), and multiplying by P−2, gives the variance of the selection gradient,
The integral is the curvature of the Malthusian fitness function averaged over the phenotype distribution in the average environment, γ (Lande and Arnold 1983). This establishes that around an optimum phenotype Var [β(t)] is approximately constant, independent of the mean phenotype, as is the infinitesimal variance of the mean phenotype (eq. 7c).
This example can be made more precise by assuming that the Malthusian fitness function is quadratic, and that θ(t) has a normal distribution through time. The Appendix shows that this adds a constant quantity to the environmental covariance between phenotypes and hence to σ2e, which therefore does not vanish when β= 0 as suggested by (13b). The same constant becomes subtracted from the expected relative fitnesses and, as remarked after (5), does not influence selection, leaving (13c) unchanged.
The asymptotic distribution of the mean phenotype, , around an optimum is given by Wright's (1931, 1969) stationary distribution. Using the gradient form of the infinitesimal mean, M (eq. 10c), with constant infinitesimal variance, V (eqs. 7c, 13c), this has the shape . For the distribution of the mean phenotype to be concentrated around a maximum of therefore requires either a low heritability, G/P, or a small variance of the standardized selection gradient or standardized selection differential, P Var [β(t)]= Var [S(t)]/P (see after eq. 7c). The heritability never exceeds unity (Falconer and MacKay 1996) and P Var [β(t)] will be small unless selection fluctuates intensely (Slatkin and Lande 1976; Bull 1987; Lande and Shannon 1996). Thus the condition often will be satisfied.
Figure 2 depicts a numerical example of the stochastic maximization of the long-run growth rate of a population by phenotypic evolution.
The single locus haploid model and the quantitative character model of evolution in a random environment share some important features, although differing greatly in their dynamics due to different modes of inheritance. In particular, they have analogous forms for the expected relative fitness of a genotype or phenotype, and for the expected selection gradient.
The results here show that although the expected evolution maximizes the long-run growth rate of a population, for a genotype or phenotype within a population the long-run growth rate is not a valid measure of fitness. The haploid model and the quantitative character model agree that the expected relative fitness of an individual is the Malthusian fitness of its genotype or phenotype in the average environment minus the covariance of its growth rate with that of the population (eqs. 4a, 10a). Fluctuating selection therefore causes the expected relative fitness to be frequency dependent. Under frequency-dependent selection in a constant environment individual relative fitnesses change with time, such that the mean fitness usually is not maximized (Wright 1949, 1969). Thus, it is remarkable that an unchanging adaptive topography exists in a fluctuating environment, despite that selection is continually changing and the expected relative fitnesses of individuals are frequency dependent.
ADAPTIVE TOPOGRAPHY OF FLUCTUATING SELECTION
In both models of inheritance, starting from a given state, the expected evolution of gene frequency, p, or mean phenotype, , is proportional to a measure of genetic variability [the heterozygosity or the additive genetic (co)variance (matrix)] and the expected selection gradient, the derivative of the long-run growth rate of a population with respect to p or . The gradient form of the expected selection implies that the expected evolution increases the long-run growth rate of a population, , as a function of p in the haploid model or in the quantitative character model. The long-run growth rate of a population therefore constitutes an adaptive topography for fluctuating natural selection similar to the surface of mean fitness in the adaptive topography for gene frequencies (Wright 1931, 1969) or that for the mean phenotype (Lande 1976, 1979, 1982) in a constant environment.
The long-run growth rate of a population has two components, , where r is the population growth rate or mean Malthusian fitness in the average environment, and σ2e is the environmental variance in population growth rate, the mean environmental covariance among genotypes or phenotypes (eqs. 3 and 6). Because σ2e depends on environmental covariances among the types, the long-run growth rate of a polymorphic population generally exceeds the mean long-run growth rate of the types, and may exceed the long-run growth rate of any type (Fig. 1). In the haploid model the mean Malthusian fitness r is linear and σ2e is quadratic in the gene frequency so an intermediate maximum in depends entirely on environmental stochasticity. In the quantitative character model the mean Malthusian fitness r generally is nonlinear, with one or more local maxima produced by stabilizing selection for intermediate optimum phenotype(s) in the average environment.
Most authors believe that in a fluctuating environment an adaptive topography does not exist or would be continually changing so that a maximization of fitness can not occur (Fisher 1958, chap. 2; Crow and Kimura 1970, p. 236; Gillespie 1991, p. 305; Caswell 2001, p. 437). The present theory reveals that with a stationary distribution of environmental states the surface , as a function of gene frequency or mean phenotype, remains constant in time although selection is continually fluctuating. At any point on this surface, the selection gradient oriented in the steepest uphill direction, in conjunction with genetic variability, determines the expected rate and direction of evolution (eqs. 4a and 10c). Fluctuating selection causes stochastic perturbations around the expected evolutionary trajectory determined by the expected selection gradient, the gradient of the long-run growth rate of the population.
This maximum principle is not realized in a haploid population because the expected stabilizing selection is overwhelmed by stochasticity (Fig. 1). This occurs for two reasons. First, both the curvature of the adaptive topography, , necessary for existence of an intermediate optimum gene frequency, and the stochasticity of selection on the gene frequency are comparable in magnitude (see Appendix). Second, as the gene frequency approaches 0 or 1, the rate of evolution diminishes and, neglecting finite population size, alleles are never actually fixed or lost, but undergo “quasi-fixation” of the one with the largest long-run growth rate (Kimura 1954; Dempster 1955; Gillespie 1991). Although not defining the expected relative fitness, Frank and Slatkin (1990) observed for the haploid model in a fluctuating environment that selection is frequency-dependent and favors rare alleles, but this is overcome by stochasticity.
The evolution of quantitative characters differs substantially because, on a suitable scale of measurement, phenotypes and breeding values often are approximately normally distributed with genetic and phenotypic variability maintained nearly constant during evolution of the mean phenotype (Wright 1968, chap. 15, 1977, chaps. 7 and 8; Falconer and MacKay 1996, chaps. 11 and 12). Furthermore, deterministic stabilizing selection toward an intermediate optimum phenotype often occurs through the mean Malthusian fitness in the average environment (Lande 1982). Both of these factors facilitate a stochastic maximization of the long-run growth rate of a population by evolution of the mean phenotype (Fig. 2). Under low heritability or weakly fluctuating selection the mean phenotype achieves a quasi-stationary distribution concentrated around the optimum phenotype in the average environment (eq. 13c and after; Slatkin and Lande 1976; Bull 1987; Lande and Shannon 1996). The variance of the standardized selection gradient often may be small, as indicated by observed time series of selection on quantitative characters (Weis et al. 1992; Kruuk Merilä & Sheldon 2001; Grant and Grant 2002; Sheldon et al. 2003; Garant et al. 2004; T. Coulson, pers. comm.; B.-E. Sæther, pers. comm.). These findings accord with the view of Wright (1948, 1969, pp. 372–377) who considered fluctuating selection and random genetic drift as stochastic factors creating a stationary distribution of gene frequencies around their deterministic equilibrium values.
I thank T. Coulson, S. Engen, and B.-E. Sæther for stimulating discussions, and two reviewers for comments on the manuscript. This work was supported by a grant from the National Science Foundation, and partially conducted during a visit to the Population Biology Center at NTNU supported by a grant from the Norwegian Research Council.
The example of a fluctuating optimum phenotype can be made more precise by assuming that the Malthusian fitness function is quadratic, and that the optimum phenotype, θ(t), has a normal distribution through time, with mean and variance σ2θ.
First consider the influence of developmental plasticity by writing the mean phenotype at time t as where is mean breeding value, or the mean phenotype measured in a common reference environment (e.g., the average environment) and includes the plastic response to environmental change. For example, suppose that developmental plasticity in response to the environment is described as a simple linear function of the optimum phenotype, ] where c is a constant. This contrasts with the usual assumption of made for a constant environment. At a given time the evolutionary response of the mean breeding value, or the mean phenotype measured in the average environment, is (cf. eq. 7a) with the selection gradient
in which is the environmental noise in the optimum phenotype. This shows that partially adaptive developmental plasticity, with 0 < c < 1, effectively reduces the variance of the optimum phenotype to (1 −c)2σ2θ. For simplicity, throughout this paper I therefore analyze evolution of the mean breeding value in a common reference environment (e.g., the average environment) and view developmental plasticity as already incorporated into the effective variance of fluctuating selection.
In the average environment the Malthusian fitness of phenotype z is
The mean Malthusian fitness in a population with a given mean phenotype in the average environment is
The covariance of Malthusian fitness between phenotypes can be obtained by Taylor expansions of r(z, t) around two arbitrary phenotypes, y and z, in the average environment, and using the fourth moment of the normal distribution of θ(t) (Kendall and Stuart 1977, p. 62),
This produces the expected relative fitness of phenotype z (eq. 10a),
The environmental variance in the population growth rate, given by the mean environmental covariance of growth rates between phenotypes (eq. 6a),
does not vanish when the mean phenotype is at the optimum in the average environment as suggested by equation (13b).
The expected selection gradient for a given mean phenotype,